The Projector Method: Theory and Examples

Our work has explored an innovative way of numerically analyzing lattice gauge theories. Previously, most particle theorists have used the Monte Carlo technique introduced by Creutz. However, results obtained using this method are not definitive because of inherent difficulties in the method (stringent computation requirements and relative inefficiency). Our techniques promise potentially more accurate measurements of physically relevant quantities in these same models. Additionally we will clarify the relationship of our technique with the standard one. We developed the Projector Method technique so it would take advantage of certain simplifying features of gauge theory models. Simultaneous and independent efforts to analyze models in atomic, molecular and condensed matter physics led people to develop the Green Function Monte Carlo. Differences in formalism between our work and the latter line of research have obscured their similarities. We believe these techniques are different applications of one underlying idea. Unifying these approaches may lead to a more complete understanding of the numerical methods and ultimately, the physical models. The Projector Method possesses several advantages over the usual Monte Carlo algorithm. Since the Projector starts from the Hamiltonian and Hilbert space structure of the problem, the lattice for realistic models is three dimensional. Because of the way that samples are generated in the Projector Method, the critical slowing associated with the standard method is avoided. Finally, the presence of certain conservation laws in the Hamiltonian formulation allow the extraction of string tension between charges that are separated by a distance equal to the size of the lattice. Starting from a very general notion of what the Projector Method is, we begin applying the techniques to several model problems in chapters 2 and 3. After these examples have traced the development of the actual algorithm from the general principles of the Projector Method, a direct comparison between the Projector and Euclidean Monte Carlo is made in chapter 4. Following this is a discussion of the application of Periodic Quantum Electrodynamics in 2 and 3 spatial dimensions. Several methods for improving the efficiency of the Projector in various circumstances are outlined in the final chapter.